Theorem adjt 9852

Description: Property of an adjoint Hilbert space operator.

Assertion

Ref	Expression
adjt	|- ((T e. dom adjh /\ A e. H~ /\ B e. H~) -> (A .ih (T` B)) = (((adjh` T)` A) .ih B))

Proof of Theorem adjt

Step	Hyp	Ref	Expression
1		opreq1 3974	. . . . 5 |- (x = A -> (x .ih (T` y)) = (A .ih (T` y)))
2		fveq2 3730	. . . . . 6 |- (x = A -> ((adjh` T)` x) = ((adjh` T)` A))
3	2	opreq1d 3981	. . . . 5 |- (x = A -> (((adjh` T)` x) .ih y) = (((adjh` T)` A) .ih y))
4	1, 3	eqeq12d 1492	. . . 4 |- (x = A -> ((x .ih (T` y)) = (((adjh` T)` x) .ih y) <-> (A .ih (T` y)) = (((adjh` T)` A) .ih y)))
5		fveq2 3730	. . . . . 6 |- (y = B -> (T` y) = (T` B))
6	5	opreq2d 3982	. . . . 5 |- (y = B -> (A .ih (T` y)) = (A .ih (T` B)))
7		opreq2 3975	. . . . 5 |- (y = B -> (((adjh` T)` A) .ih y) = (((adjh` T)` A) .ih B))
8	6, 7	eqeq12d 1492	. . . 4 |- (y = B -> ((A .ih (T` y)) = (((adjh` T)` A) .ih y) <-> (A .ih (T` B)) = (((adjh` T)` A) .ih B)))
9	4, 8	rcla42v 1883	. . 3 |- ((A e. H~ /\ B e. H~) -> (A.x e. H~ A.y e. H~ (x .ih (T` y)) = (((adjh` T)` x) .ih y) -> (A .ih (T` B)) = (((adjh` T)` A) .ih B)))
10		funadj 9808	. . . . . . 7 |- Fun adjh
11		funfvop 3809	. . . . . . 7 |- ((Fun adjh /\ T e. dom adjh) -> <.T, (adjh` T)>. e. adjh)
12	10, 11	mpan 697	. . . . . 6 |- (T e. dom adjh -> <.T, (adjh` T)>. e. adjh)
13		dfadj2 9807	. . . . . 6 |- adjh = {<.z, w>. | (z:H~-->H~ /\ w:H~-->H~ /\ A.x e. H~ A.y e. H~ (x .ih (z` y)) = ((w` x) .ih y))}
14	12, 13	syl6eleq 1561	. . . . 5 |- (T e. dom adjh -> <.T, (adjh` T)>. e. {<.z, w>. | (z:H~-->H~ /\ w:H~-->H~ /\ A.x e. H~ A.y e. H~ (x .ih (z` y)) = ((w` x) .ih y))})
15		fvex 3738	. . . . . 6 |- (adjh` T) e. V
16		feq1 3626	. . . . . . . 8 |- (z = T -> (z:H~-->H~ <-> T:H~-->H~))
17		fveq1 3729	. . . . . . . . . . 11 |- (z = T -> (z` y) = (T` y))
18	17	opreq2d 3982	. . . . . . . . . 10 |- (z = T -> (x .ih (z` y)) = (x .ih (T` y)))
19	18	eqeq1d 1486	. . . . . . . . 9 |- (z = T -> ((x .ih (z` y)) = ((w` x) .ih y) <-> (x .ih (T` y)) = ((w` x) .ih y)))
20	19	2ralbidv 1683	. . . . . . . 8 |- (z = T -> (A.x e. H~ A.y e. H~ (x .ih (z` y)) = ((w` x) .ih y) <-> A.x e. H~ A.y e. H~ (x .ih (T` y)) = ((w` x) .ih y)))
21	16, 20	3anbi13d 897	. . . . . . 7 |- (z = T -> ((z:H~-->H~ /\ w:H~-->H~ /\ A.x e. H~ A.y e. H~ (x .ih (z` y)) = ((w` x) .ih y)) <-> (T:H~-->H~ /\ w:H~-->H~ /\ A.x e. H~ A.y e. H~ (x .ih (T` y)) = ((w` x) .ih y))))
22		feq1 3626	. . . . . . . 8 |- (w = (adjh` T) -> (w:H~-->H~ <-> (adjh` T):H~-->H~))
23		fveq1 3729	. . . . . . . . . . 11 |- (w = (adjh` T) -> (w` x) = ((adjh` T)` x))
24	23	opreq1d 3981	. . . . . . . . . 10 |- (w = (adjh` T) -> ((w` x) .ih y) = (((adjh` T)` x) .ih y))
25	24	eqeq2d 1489	. . . . . . . . 9 |- (w = (adjh` T) -> ((x .ih (T` y)) = ((w` x) .ih y) <-> (x .ih (T` y)) = (((adjh` T)` x) .ih y)))
26	25	2ralbidv 1683	. . . . . . . 8 |- (w = (adjh` T) -> (A.x e. H~ A.y e. H~ (x .ih (T` y)) = ((w` x) .ih y) <-> A.x e. H~ A.y e. H~ (x .ih (T` y)) = (((adjh` T)` x) .ih y)))
27	22, 26	3anbi23d 898	. . . . . . 7 |- (w = (adjh` T) -> ((T:H~-->H~ /\ w:H~-->H~ /\ A.x e. H~ A.y e. H~ (x .ih (T` y)) = ((w` x) .ih y)) <-> (T:H~-->H~ /\ (adjh` T):H~-->H~ /\ A.x e. H~ A.y e. H~ (x .ih (T` y)) = (((adjh` T)` x) .ih y))))
28	21, 27	opelopabg 2823	. . . . . 6 |- ((T e. dom adjh /\ (adjh` T) e. V) -> (<.T, (adjh` T)>. e. {<.z, w>. | (z:H~-->H~ /\ w:H~-->H~ /\ A.x e. H~ A.y e. H~ (x .ih (z` y)) = ((w` x) .ih y))} <-> (T:H~-->H~ /\ (adjh` T):H~-->H~ /\ A.x e. H~ A.y e. H~ (x .ih (T` y)) = (((adjh` T)` x) .ih y))))
29	15, 28	mpan2 698	. . . . 5 |- (T e. dom adjh -> (<.T, (adjh` T)>. e. {<.z, w>. | (z:H~-->H~ /\ w:H~-->H~ /\ A.x e. H~ A.y e. H~ (x .ih (z` y)) = ((w` x) .ih y))} <-> (T:H~-->H~ /\ (adjh` T):H~-->H~ /\ A.x e. H~ A.y e. H~ (x .ih (T` y)) = (((adjh` T)` x) .ih y))))
30	14, 29	mpbid 195	. . . 4 |- (T e. dom adjh -> (T:H~-->H~ /\ (adjh` T):H~-->H~ /\ A.x e. H~ A.y e. H~ (x .ih (T` y)) = (((adjh` T)` x) .ih y)))
31	30	3simp3d 798	. . 3 |- (T e. dom adjh -> A.x e. H~ A.y e. H~ (x .ih (T` y)) = (((adjh` T)` x) .ih y))
32	9, 31	syl5com 52	. 2 |- (T e. dom adjh -> ((A e. H~ /\ B e. H~) -> (A .ih (T` B)) = (((adjh` T)` A) .ih B)))
33	32	3impib 833	1 |- ((T e. dom adjh /\ A e. H~ /\ B e. H~) -> (A .ih (T` B)) = (((adjh` T)` A) .ih B))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960  A.wral 1648  Vcvv 1814  <.cop 2415  {copab 2671  dom cdm 3176  Fun wfun 3182  -->wf 3184  ` cfv 3188  (class class class)co 3969  H~chil 8783   .ih csp 8788  adj_hcado 8819

This theorem is referenced by:  adj2t 9853  adjadjt 9855  hmopadj2t 9860

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634  ax-hfvadd 8865  ax-hvcom 8866  ax-hvass 8867  ax-hv0cl 8868  ax-hvaddid 8869  ax-hfvmul 8870  ax-hvmulid 8871  ax-hvdistr2 8874  ax-hvmul0 8875  ax-hfi 8941  ax-his1 8944  ax-his2 8945  ax-his3 8946  ax-his4 8947

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-nel 1591  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-en 4374  df-dom 4375  df-sdom 4376  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-1p 5099  df-plp 5100  df-mp 5101  df-ltp 5102  df-plpr 5176  df-mpr 5177  df-enr 5178  df-nr 5179  df-plr 5180  df-mr 5181  df-ltr 5182  df-0r 5183  df-1r 5184  df-m1r 5185  df-c 5252  df-0 5253  df-1 5254  df-i 5255  df-r 5256  df-plus 5257  df-mul 5258  df-lt 5259  df-sub 5368  df-neg 5370  df-pnf 5499  df-mnf 5500  df-xr 5501  df-ltxr 5502  df-le 5503  df-div 5715  df-re 6752  df-im 6753  df-cj 6754  df-hvsub 8835  df-adjh 9770

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